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lecture note 4

# lecture note 4 - C2M2 Rational Fractions or Partial...

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C2M2 Rational Fractions or Partial Fraction Decompositions When you were learning basic algebra there were probably problems assigned on adding fractions which had constants, linear expressions, and quadratic expressions in x in the numerators and denominators. Our objective here is to take the answers to those questions and find the fractions that were added together. The need here is to break down a complicated fraction into several simple ones whose anti-derivatives are (much!) easier to find. To approach this systematically we separate the problems into groups determined by the nature of the denominators of the fractions. Recall that P ( x ) = x 2 a 2 can be factored into P ( x ) = ( x a )( x + a ), so we say that P ( x ) is reducible . Likewise, Q ( x ) = x 2 + a 2 can NOT be factored, so Q ( x ) is irreducible . There are different approaches to to these problems and we will use substitution as our method. Two principles from algebra are applicable here. The first states that if two polynomials in x agree for all values of x , then the polynomials have the same coeﬃcients, that is, they are identical. The second states that x r is a factor of the polynomial P ( x ) if and only if P ( r ) = 0.

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